Tropical polyhedra: Types and basic properties

[ooinaruhugh]

Polymake has support for tropical polyhedra, but what is there does not interface so nicely with OSCAR.
This PR serves to add a proper Julia type TropicalPolyhedron and provide basic functionality, to fix some design decisions about the data type.

This might also be useful in light of #3989 (review).

EDIT: I discussed this PR with @micjoswig and we came up with the following outline.

• Given a set of n points in tropical projective d-space, we want to compute the induced covector decomposition of the tropical projective space.
• For tropical polytopes with only finite coordinates, the covector decomposition yields a polyhedral subdivision of the tropical projective torus. This is implemented in polymake and should be taken from there.
• With infinite coordinates, we obtain an induced covector decomposition in each stratum (which is a tropical projective torus of suitable dimension).
• For arbitrary point sets, the cells in this decomposition might live in different strata of tropical projective space. In each stratum, this gives a polyhedral complex, which all fit together as a cell complex.
• Thus, a data type for a tropical polytope needs keep track of data (e.g. PolyhedralComplexes) for each stratum.

In terms of data types: try a map which send a set (describing the stratum) to a polyhedral complex. Beware that polyhedral complexes require the vertices to be numbered consecutively. Because each stratum only sees a subset of the given points, we need an array and a map for translating indices back and forth.

[ooinaruhugh]

@Sami-Halaseh

[codecov]

Codecov Report

Attention: Patch coverage is 59.04762% with 43 lines in your changes missing coverage. Please review.

Project coverage is 84.82%. Comparing base (5070b99) to head (6f3d94e).
Report is 3 commits behind head on master.

Files with missing lines	Patch %	Lines
.../TropicalGeometry/TropicalPolyhedron/properties.jl	61.64%	28 Missing ⚠️
...ropicalGeometry/TropicalPolyhedron/constructors.jl	53.57%	13 Missing ⚠️
src/TropicalGeometry/semiring.jl	50.00%	2 Missing ⚠️

Additional details and impacted files

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##           master    #4001      +/-   ##
==========================================
- Coverage   84.88%   84.82%   -0.07%     
==========================================
  Files         697      699       +2     
  Lines       94180    94337     +157     
==========================================
+ Hits        79948    80019      +71     
- Misses      14232    14318      +86     

Files with missing lines	Coverage Δ
src/TropicalGeometry/semiring.jl	63.50% <50.00%> (-0.41%)	⬇️
...ropicalGeometry/TropicalPolyhedron/constructors.jl	53.57% <53.57%> (ø)
.../TropicalGeometry/TropicalPolyhedron/properties.jl	61.64% <61.64%> (ø)

... and 8 files with indirect coverage changes

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[ooinaruhugh]

At the moment, questions that need to be discussed:

• How to handle the bookkeeping accross the strata? (@lkastner Might have an opinion, e.g. the question Map vs. indexing strata by bitmasks)
• Regarding the "one polyhedral complex per stratum", vertices need to be renamed. Where to do this, or how to present information on boundary cells?
• More remote question, this is also a welcome addition for polymake. Thus, implement in polymake first and then make accessible from OSCAR, or directly in OSCAR?

[fingolfin]

What the status of this PR?

[ooinaruhugh]

This has been pushed to my backlog over the last few months, but I just finished a great deal of work. So thank you for reminding me of this.

At the moment, I just had some prototype code.

[micjoswig]

Let's first finalize #4061. Afterwards it will be easier to see how we want to design this.

One aspect to take into account is that @YueRen chose a nonhomogeneous setting. However, the natural interface between tropical convexity and tropical algebraic geometry goes through tropical hyperplane arrangements described by homogeneous linear forms.

[fingolfin]

In the meantime #4061 has been merged a few weeks ago. So if @ooinaruhugh is still interested and has some spare time, perhaps this can be brought forward? :-)

[micjoswig]

The overall goal still stands.

To make progress now, we tackle the following slightly simpler problem. first Input: point configuration in tropical projective space; Output: covector decomposition induced on tropical projective torus (which is an ordinary polyhedral complex). This can be inherited from polymake directly.

[ooinaruhugh]

While testing the polytope_covector_decomposition, I noticed inconsistencies regarding what 'pseudovertices' in polymake are. PSEUDOVERTICES from polymake gives rays for some reason, and POLYTOPE_COVECTOR_DECOMPOSITION does not agree with the visualization of the tropical polytope (from VISUAL).
We need to investigate this on the polymake side.

[benlorenz]

While testing the polytope_covector_decomposition, I noticed inconsistencies regarding what 'pseudovertices' in polymake are. PSEUDOVERTICES from polymake gives rays for some reason, and POLYTOPE_COVECTOR_DECOMPOSITION does not agree with the visualization of the tropical polytope (from VISUAL). We need to investigate this on the polymake side.

Can you provide an explicit (polymake) example of this discrepancy?

[fingolfin]

ping @ooinaruhugh

[ooinaruhugh]

The discussed discrepancies meant that we had to fix some parts in polymake regarding tropical polytopes. I'll finish the test cases which reflect how the output of the implemented functions should be mathematically speaking. But we have to wait for v4.14 of polymake in the end. Unless I should implement a temporary workaround that's only going to stay for a short time.

[fingolfin]

Would be nice to get this merged. There is a bunch of minor code review comments that could be addressed with a few clicks of "apply suggestions". A couple others seem to be resolved and just need to be marked as such.

There is also the issue of the "polymake hotfix", no idea what the status there is?

[benlorenz]

There is also the issue of the "polymake hotfix", no idea what the status there is?

It was released as part of polymake 4.15 and is available as polymake_jll-400.1500.000. This version is already in use by Oscar.
Strictly speaking we could either add a compat entry for polymake_jll to enforce this, or set broken= pmversion < v"400.1500.0" but I don't think this is necessary since there is nothing else that could block this update.

[YueRen]

@ooinaruhugh Looks great, thanks for your contribution! Here just some very minor nitpicks.

[fingolfin]

Looks good to me, thank you very much!

[YueRen]

Strong approve.